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Mirrors > Home > ILE Home > Th. List > m1expeven | GIF version |
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
Ref | Expression |
---|---|
m1expeven | ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9256 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | 2timesd 9159 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) = (𝑁 + 𝑁)) |
3 | 2 | oveq2d 5890 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁))) |
4 | neg1cn 9022 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | neg1ap0 9026 | . . . 4 ⊢ -1 # 0 | |
6 | expaddzap 10561 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 # 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) | |
7 | 4, 5, 6 | mpanl12 436 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
8 | 7 | anidms 397 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
9 | m1expcl2 10539 | . . 3 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | |
10 | neg1rr 9023 | . . . . . 6 ⊢ -1 ∈ ℝ | |
11 | reexpclzap 10537 | . . . . . 6 ⊢ ((-1 ∈ ℝ ∧ -1 # 0 ∧ 𝑁 ∈ ℤ) → (-1↑𝑁) ∈ ℝ) | |
12 | 10, 5, 11 | mp3an12 1327 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℝ) |
13 | elprg 3612 | . . . . 5 ⊢ ((-1↑𝑁) ∈ ℝ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) |
15 | oveq12 5883 | . . . . . . 7 ⊢ (((-1↑𝑁) = -1 ∧ (-1↑𝑁) = -1) → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) | |
16 | 15 | anidms 397 | . . . . . 6 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) |
17 | neg1mulneg1e1 9129 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
18 | 16, 17 | eqtrdi 2226 | . . . . 5 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
19 | oveq12 5883 | . . . . . . 7 ⊢ (((-1↑𝑁) = 1 ∧ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) | |
20 | 19 | anidms 397 | . . . . . 6 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) |
21 | 1t1e1 9069 | . . . . . 6 ⊢ (1 · 1) = 1 | |
22 | 20, 21 | eqtrdi 2226 | . . . . 5 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
23 | 18, 22 | jaoi 716 | . . . 4 ⊢ (((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
24 | 14, 23 | syl6bi 163 | . . 3 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} → ((-1↑𝑁) · (-1↑𝑁)) = 1)) |
25 | 9, 24 | mpd 13 | . 2 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
26 | 3, 8, 25 | 3eqtrd 2214 | 1 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 {cpr 3593 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 -cneg 8127 # cap 8536 2c2 8968 ℤcz 9251 ↑cexp 10516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-n0 9175 df-z 9252 df-uz 9527 df-seqfrec 10443 df-exp 10517 |
This theorem is referenced by: m1expe 11898 m1expo 11899 m1exp1 11900 |
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